Question

Carry out the integration indicated in equation \((10-10)\).

Short answer

The result of the integral \(\int (10 -10) \, dx\) is \(0x + c\).

Step-by-step solution

Evaluate the subtraction

Initially, simply calculate the subtractions \(10 - 10 = 0\).

Integral of zero

Now, reintegrate \(0\).The integral of \(0\) is then simply \(0x + c\) where c is the constant of integration.

Key concepts

Calculus in Physics

Calculus is the mathematical study of continuous change, and it constitutes one of the central branches in mathematics. In the realm of physics, calculus is indispensable as it provides the tools for modeling and solving problems involving the motion of particles, forces, and energy. Take the motion of a planet orbiting a star, for example. Calculus helps physicists understand this motion by letting them calculate velocities and accelerations from the planet's position.

For instance, differentiation in calculus is used to determine the velocity, which is the rate of change of position with respect to time. On the other hand, integration in physics is often utilized to find the distance traveled by an object when its velocity as a function of time is known. Therefore, calculus enables physicists to move seamlessly between these different quantities, each providing insights into physical phenomena.

Constant of Integration

In calculus, the constant of integration, usually denoted as 'c', arises naturally when performing indefinite integrals (antiderivatives). When the derivative of a function is taken, detailed information about its original constant terms is lost. To account for this, when the integral is calculated, a 'c' is added to represent any constant value that could have been present in the original function.

In physics, this becomes particularly important because the constant of integration can often represent initial conditions or physical constants within a given problem. For example, when dealing with kinematics equations, the constant might represent the initial position or velocity of an object. The constant ensures the solution to the integral is a family of functions that encompass all possible scenarios the physical system might be in.

Integrals in Modern Physics

The application of integrals extends far beyond classical mechanics and into modern physics, including fields such as quantum mechanics, electromagnetism, and general relativity. In electromagnetism, for example, integrals are used to calculate quantities like electric potential and magnetic field strength over continuous charge distributions.

In quantum mechanics, the wavefunction of a particle is governed by the Schrödinger equation, where integrals are involved in finding the probabilities of a particle's position and momentum. Meanwhile, in general relativity, integrals help in understanding the curvature of spacetime caused by mass and energy. The importance of integrals in modern physics lies in their ability to handle complex systems where discrete sums are not practical and continuous distributions of properties must be considered.